The optimal sample sizes of the Xbar and CUSUM charts

This article studies the sample sizes of two most commonly used control charts, the X¿ chart and the CUSUM chart. It takes the sampling cost (including the variable and fixed components) into consideration and uses a quadratic loss function as a performance measure. It is interesting to find that if n_X¿ = 4 and n_CUSUM = 1, the X¿ chart often outperforms the CUSUM chart. Furthermore, the overall performance of both charts can be improved by an optimization design using the quadratic loss function as the objective. The optimal sample size depends on the range of mean shift ¿ and the ratio between the fixed and variable sampling costs. For general cases (0 < ¿ ¿ 4), the best sample sizes are n_X¿ = 3 for the X¿ chart and n_CUSUM = 2 or 3 for the CUSUM chart.